In this paper, we are concerned with the problem of counting Euler tours on
graphs of bounded treewidth. Many problems which are \(\mbox{{\sf NP}}\) -hard for general
graphs, can be solved in polynomial time on bounded treewidth graphs. Indeed, a
result of Courcelle [1]} asserts that any graph property that is expressible
in Monadic Second Order logic (with edge quantifiers) can be solved in linear time on
bounded treewidth graphs. Elberfeld et al. [2]} adapt the theorem of Courcelle in
the parallel setting and prove a L bound. However, Eulerianity is provably
not \({\sf MSO}\)  -expressible [3]} and hence the approaches mentioned above are not directly
applicable in our context.
